## Abstract

Nonlinear nanophotonics is a rapidly developing field of research with many potential applications for the design of nonlinear nanoantennas, light sources, nanolasers, and ultrafast miniature metadevices. A tight confinement of the local electromagnetic fields in resonant photonic nanostructures can boost nonlinear optical effects, thus offering versatile opportunities for the subwavelength control of light. To achieve the desired functionalities, it is essential to gain flexible control over the near- and far-field properties of nanostructures. To engineer nonlinear scattering from resonant nanoscale elements, both modal and multipolar control of the nonlinear response are widely exploited for enhancing the near-field interaction and optimizing the radiation directionality. Motivated by the recent progress of all-dielectric nanophotonics, where the electric and magnetic multipolar contributions may become comparable, here we review the advances in the recently emerged field of multipolar nonlinear nanophotonics, starting from earlier relevant studies of metallic and metal–dielectric structures supporting localized plasmonic resonances to then discussing the latest results for all-dielectric nanostructures driven by Mie-type multipolar resonances and optically induced magnetic response. These recent developments suggest intriguing opportunities for a design of nonlinear subwavelength light sources with reconfigurable radiation characteristics and engineering large effective optical nonlinearities at the nanoscale, which could have important implications for novel nonlinear photonic devices operating beyond the diffraction limit.

© 2016 Optical Society of America

## 1. INTRODUCTION

Modern nanophotonics aims toward efficient light manipulation at the nanoscale and the design of ultrafast compact optical devices for fully functional photonic circuitry integrable with state-of-art nanoscale electronics [1–6]. After decades of fruitful developments, nanophotonics has become a prominent area of research, with applications ranging from high-performance data processing and optical computing to super-imaging and biosensing [7–12]. Many branches of nanophotonics commonly rely on strong light–matter interaction in the resonant subwavelength structures due to the enhanced near fields [13], achieved through the excitation of trapped electromagnetic modes sustained by the nanoelements, or localized geometrical resonances [14]. Resonant tight field confinement is indispensable particularly for efficient intensity-dependent effects, with practically important nonlinear optical applications such as frequency conversion, wave mixing, Raman scattering, self-action, and all-optical switching. While in bulk macroscopic media efficiencies of nonlinear optical phenomena are largely determined by the intrinsic nonlinear properties and phase-matching requirements [15], efficiencies of nonlinear processes and effective susceptibilities, which may be assigned to nanostructured materials to characterize their nonlinear response, can be significantly increased through enhanced local fields at resonances. Therefore, nonlinear response significantly depends on localized resonant effects in nanostructures, allowing for nonlinear optical components to be scaled down in size and offering exclusive prospects for engineering fast and strong optical nonlinearity and all-optical light control at the nanoscale.

Present-day nanophotonics has greatly advanced in the manufacturing and study of open nanosize optical resonators that are actually resonant *nanoantennas*, which couple localized electromagnetic fields and freely propagating radiation [16,17]. Metal nanoparticles [14] sustaining surface plasmon-polariton modes as well as dielectric counterparts with high enough index of refraction [18,19] are utilized as such resonators, and serve for applications in nonlinear diagnostics and microscopy. In turn, two- and three-dimensional clusters and regular arrays of such nanoparticles, employed as unit cells resonantly responding to IR radiation and visible light, constitute optical metasurfaces and metamaterials (artificial media with pre-designed properties) by analogy with microwave systems [20–22].

*Multipole decomposition*. Generally, in the problems of both linear and nonlinear scattering at arbitrary nanoscale objects, multipole decomposition of the scattered electromagnetic fields provides a transparent interpretation for the measurable far-field characteristics, such as radiation efficiency and radiation patterns, since they are essentially determined by the interference of dominating excited multipole modes [23–30].

In terms of the electric ${a}_{E}$ and magnetic ${a}_{M}$ scattering coefficients, the total time-averaged scattered power (energy flow) is given by

Assuming the far-field asymptotic of the outgoing spherical wave, the field expansion into the vector spherical harmonics ${\mathbf{X}}_{l,m}(\theta ,\phi )$ recovers the directional dependence of the radiation,

*unidirectional scattering*[40] from single-element antennas. The scattering is interferencially suppressed in either the backward or forward ($z$) direction, if the relation $1/{\u03f5}_{0}{p}_{x}=\pm \eta {m}_{y}$ holds for the only non-negligible Cartesian electric ${p}_{x}$ and magnetic ${m}_{y}$ dipolar moments induced in a particle along the $x$ and $y$ axes, respectively. Developing this concept, the directionality of scattering can be improved through the interference of properly excited higher-order electric and magnetic multipolar modes relying on both amplitude and phase engineering (the so-called generalized Kerker scattering) [41–44].

However, the local-field features and far-field properties are intimately linked in the nonlinear response of nanostructures [45,46]. A formal way to quantify this effect is based on the Lorentz reciprocity theorem. In particular, this method allows one to predict the metamaterial nonlinearity by using linear calculations [46], and to express the excitation coefficients of the spherical multipoles the radiated field is decomposed into, through the overlap integrals of the nonlinear source and the respective spherical modes [47]. In this regard, two key ingredients to realizing strong nonlinear response from nanostructures are usually emphasized: the local field enhancement and modal overlaps. For instance, the efficiency of harmonic generation can be strongly enhanced in nanostructures, provided the pump or generated frequency matches the supported resonance [15], especially if the geometry is doubly resonant [48–52], i.e., it sustains resonances at both the fundamental and harmonic frequencies, and spatial distribution of the nonlinear source is such that it strongly couples the corresponding modes.

For metamaterials, the applicability of simple estimates based on the nonlinear oscillator model [53,54] and Miller’s law [55,56] is limited to the specific cases only [46,57], e.g., odd in frequency third-harmonic generation [58–60]. In fact, the character and efficiency of the nonlinear response in nanostructures are highly affected by many factors [61], including variations in their size, shape, material filling fractions, quality of samples, interparticle interactions, spatial symmetries of both geometry and excitation beams [62], and linear scattering properties. Nonetheless, to a large extent, the key governing features may be approached with the analysis of the unique resonant behavior of these artificial materials [63], as well as strengths and mutual interference of the leading excited or generated field multipoles [64].

*Nonlinear response*. Within the macroscopic description utilizing Maxwell’s equations, nonlinear optics employs the nonlinear constitutive relations. For example, in the electric dipole approximation of light–matter interaction, the material response in a nonmagnetic medium can be specified by the nonlinear relationship between the applied electric field $\mathbf{E}$ and induced polarization $\mathbf{P}$ as

Symmetry considerations appear to be of particular importance in nonlinear optics. For instance, the second-order nonlinear effects are inhibited in the bulk of uniform centrosymmetric media such as plasmonic metals and group IV semiconductors, within the electric dipole approximation of the light–matter interaction because of the symmetry constraints, while no such restriction exists for third-order processes [65]. However, the inversion symmetry is broken at interfaces, thus enabling the second-order nonlinear processes from surfaces of isotropic crystals due to the electric-dipole surface contribution to the nonlinear polarization. Its sensitivity to surface properties is used in probing techniques. The bulk nonlinear polarization arises from higher-order nonlocal magnetic-dipole and electric-quadrupole interactions with light at the microscopic level. To account for the multipolar orders, the effective light–matter interaction Hamiltonian [1,66] is expanded as follows

Strong interaction of subwavelength structures with an external light field is most often related to resonant excitation of low-order modes (up to quadrupoles) due to (i) surface plasmon resonances in plasmonic materials such as metals and doped graphene (Section 2), or (ii) Mie-type resonances in high-permittivity dielectric nanoparticles supporting large displacement currents (Section 4). The origin of these resonances can be understood with analogy to the seminal problem of Mie scattering by spherical particles. The macroscopic nonlinear polarization source is then defined by the induced near-field distribution, and it is expressed in terms of multipolar bulk and surface contributions inherent to the nonlinear medium. Nonlinear response functions result from averaging multipolar light–matter interactions, which can be described by the Hamiltonian ${H}_{\mathrm{int}}$ at the microscopic level, with account for the symmetry and structural properties of the material. The induced nonlinear polarization may generate electric and magnetic multipoles of different orders, yielding nontrivial radiation and polarization patterns through the far-field interference. For instance, appealing to this ideology, a magnetic Mie mode can support strong fields that drive the electric-dipole-allowed bulk nonlinearity of the material and, in turn, the nonlinear source induced in the nanoparticle can give rise to higher-order multipoles in nonlinear scattering (Section 4).

The aim of this review paper is twofold. First, within the connecting frame of multipolar field decomposition and scattering, we discuss the nonlinear optical effects beyond the diffraction limit driven by interference of several multipolar contributions from both electric and magnetic resonances. Importantly, the physics of supported resonances and material properties in plasmonic and dielectric nanoparticles are different, as illustrated in Fig. 1, and therefore we consider metallic, metal–dielectric and all-dielectric structures separately. Our main focus is on the recent studies of all-dielectric structures, but we also review very briefly some relevant earlier results on multipolar effects driven by plasmonic resonances. Second, we emphasize the importance of the optically induced magnetic resonances in hybrid and mainly high-index dielectric nanostructures which may lead to a substantial enhancement and reshaping of the nonlinear response and suggest novel ways to control directionality of the nonlinear radiation.

## 2. PLASMONIC NANOPARTICLES

To date, the possibilities of nanoscale confinement of light are primarily associated with plasmon-polaritons, which are collective excitations originating from coupling of the electromagnetic fields to electron oscillations in a metal plasma [7]. Physics of light interaction with metal structures that are much smaller than the free space wavelength of light constitutes one of the most significant branches of contemporary nanophotonics, *nanoplasmonics*. Combining strong localized surface plasmon resonances and high intrinsic nonlinearities, plasmonic structures offer a unique playground to study a rich diversity of nonlinear phenomena, including SHG [8,67–71], THG [58–60,65], four-wave mixing [72], multiphoton luminescence [73], and self-action effects [74–78]. In a broader scope, the intense electric near-fields may also boost (enhance) the nonlinear response in the materials brought into the proximity of plasmonic nanoantennas [79–84]. Some important aspects of *nonlinear plasmonics* were reflected in several recent review papers [68–70,85].

Plasmons are commonly associated with conventional metals (gold, silver, copper, and aluminum) possessing a large number of quasi-free conduction electrons which oscillate collectively in response to the applied harmonic field. *Electric dipolar resonance* caused by such plasma oscillations in finite structures is most widely exploited for small metallic particles and their composites in nonlinear plasmonics [58–60,68]. Noticeably, for deep subwavelength scatterers, even when the excitation wavelength is strongly detuned from the resonance frequency, the electric dipole order dominates over the higher-order excitations in the multipolar decomposition.

The bulk of plasmonic metals formed by atoms organized in face-centered cubic lattices exhibits a center of symmetry. Assuming a particle made of a centrosymmetric isotropic homogeneous medium, the second-harmonic (SH) polarization source can be written as a sum of nonlocal-bulk and local-surface contributions,

To describe the nonlinear scattering theoretically, the nonlinear Mie theory was developed over recent decades [29,68,96–101] as an extension of the analytical approach outlined in Refs. [25,102] for the Rayleigh limit of SHG from a spherical particle. This theory can be used for particles of any size and material, though being limited to strictly spherical shapes, and it is widely exploited to examine SHG governed by the dominant surface SH polarization source [87,88] in spheroidal metallic nanoparticles. In contrast, another technique, the nonlinear Rayleigh–Gans–Debye model, is applicable to particles of any shape as long as a refractive index mismatch between the particle and a host medium is low. However, the latter is not the case for metallic particles immersed in aqueous medium, which are frequently employed in experiments. Numerical studies in nonlinear nanophotonics [103–105] are most commonly performed with the use of the finite-element method [57,106–108], integral equation techniques [109–111], hydrodynamic models [92,94], and finite-difference time-domain method [112].

In the small-particle limit $kR\ll 1$, it was proven analytically that nonlinear local electric quadrupole and nonlocal electric dipole, originating from the retardation effect, excited in a spherical metal nanoparticle of radius $R$ at the SH frequency, provide the main contributions to the SHG radiation [25,102], whereas the linear scattering properties of small particles are largely determined by the electric dipolar response. This deduction has been also supported by extensive experimental studies [108,113,114]. Deformation effects, causing the deviation of the particle shape from a perfect sphere, allow for the SH emission from the locally excited dipole [115,116]. By increasing the particle size, the onset of an octupolar contribution was detected for Au nanoparticles 70 nm in diameter [88,107].

Figure 2(a) demonstrates the angle-resolved second-harmonic scattering (AR-SHS) pattern from 80 nm diameter Ag nanoparticles in water excited at the fundamental wavelength 800 nm [117]. In agreement with theoretical predictions, the dominant surface susceptibility component ${\chi}_{\perp \perp \perp}^{(2)}$ drives a quadrupolar-like SH emission. For comparison, an AR-SHS pattern from nonplasmonic, similarly sized spherical polystyrene nanoparticles adsorbed with nonlinear optically active malachite green molecules (MG/PS) (88 nm diameter) pumped at 840 nm is shown in the inset. In the latter case, nonlocally excited dipole emission characterized by two lobes stems from the different leading component ${\chi}_{\perp \Vert \Vert}^{(2)}$.

Spectral and angular properties of SHG from plasmonic nanoparticles are largely influenced by structural symmetries of both scatterer and external field, a type of excitation [118,119], spatial variations of the applied electromagnetic field, and field gradients at nanoscale, and it can be significantly enhanced in the vicinity of plasmonic resonances known for their inherent geometric tunability. For instance, synthetic nonlinear media can be designed from reduced-symmetry 3D plasmonic elements such as nanocups consisting of a spherical metal semishell fabricated around a dielectric nanoparticle core [120]. Symmetry-breaking in this geometry introduces a magnetic dipole mode, when a nanocup resembles a resonant LC circuit. Both intensity and emission profiles of the enhanced SHG from such a nanocup excited at the magnetic dipole resonance were observed to depend on a nanocup orientation. The conversion efficiency increases dramatically as the angle between the pump excitation direction and the nanocup symmetry axis is increased, as shown in Fig. 2(b), while the scattering directionality is dictated by a nanocup inclination, regardless of the excitation direction or polarization.

In the context of plasmonic nanoantennas, the idea of all-optical control of the radiation directionality through an interplay between linear scattering and second-harmonic generation was proposed in Ref. [32]. In the proof-of-concept theoretical model, the authors consider a 100 nm gold nanowire simultaneously excited by a low-intensity (signal) and high-intensity (control) fields at wavelengths $\lambda =532\text{\hspace{0.17em}}\mathrm{nm}$ and $2\lambda $, respectively. Asymmetric scattering pattern occurs owing to the interference between the linear electric dipole induced by the signal and nonlinear dipole and quadrupoles generated by the control, as exemplified in Fig. 2(c). Altering excitation conditions, the authors demonstrate that light scattering from a single nanoelement can be efficiently tuned by changing phase delay and relative angle between two excitation beams. Similarly, in Ref. [121], it was shown that THG signal emitted from a three-nanorod gold structure can be coherently controlled by changing a relative phase of two different excitation channels. Localized surface plasmon polaritons in the nanorods are excited through near- and far-field couplings to two orthogonally polarized light fields. The cancellation of the excitation was measured as a minimum of the THG signal, and the observed phase shift between the THG signal and excitation was attributed to damping effects.

Graphene, a single atomic layer of graphite, has emerged recently as a promising alternative to noble metals for applications in plasmonics [122,123]. The study of plasmonic effects in doped graphene structures has attracted special interest from the nanoplasmonics research community due to novel functionalities suggested by such systems, including an extraordinary field confinement by a graphene layer, tunability of graphene properties through doping or electrostatic gating, and longer lifetimes in the infrared and terahertz frequency ranges [124]. In addition, graphene demonstrates strong and tunable optical nonlinearity and it can be incorporated into various components of nanoscale optics [125–127]. The study of the nonlinear response of graphene in the resonant plasmonic geometries is still in its infancy and is expected to bring new concepts and find real applications. Remarkably, graphene is not only employed for planar geometries, but there are also attempts to wrap complex three-dimensional objects in graphene [128–130]. In Ref. [36], a theoretical model for the resonant second-harmonic generation from a graphene-wrapped dielectric spherical nanoparticle has been developed. Strong nonlinear response is caused by an induced quadratic surface current in graphene. When the particle is illuminated by a plane wave, the resonantly enhanced radiation of the second harmonic is quadrupolar and symmetric; however, the radiation pattern of the second harmonic can be manipulated by placing the particle in a weakly inhomogeneous external field, which is schematically shown in Fig. 2(d). This resultant asymmetric emission is determined by the interference of dipole and quadrupole modes and the asymmetry of the external field changes the efficiency of the excitation of these modes. For the typical parameters, the second harmonic is predominantly radiated in the half-space with stronger external electric field.

A macroscopic description of the nonlinear response from arrays of noncentrosymmetric nanoparticles is based on the effective multipolar tensors [131–134]. In particular, the magnetic dipole and electric quadrupole contributions were experimentally identified in the nonlinear emission of L-shape gold nanoparticles through asymmetry of the generated SH field intensity measured in transmission and reflection [135]. The second-order nonlinear response of arrays of metal nanostructures and the relative importance of such multipolar effects are generally affected by the symmetry properties and shapes of scatterers, quality of samples, plasmonic enhancement and small-scale inhomogeneities of the local field, hot spots, particle arrangement, and interparticle coupling [136–139].

Being the simplest nonlinear optical process, SHG was intensively studied in various plasmonic structures of different shapes and coupling [137,139–146], Fano-resonant [147,148] and chiral [149–151] geometries. A structure is termed chiral if it lacks an internal plane of symmetry so that it cannot be superimposed on its mirror image. Being highly sensitive to mirror symmetry transformations, surface SHG from chiral metallic metasurfaces strongly depends on the handedness of the incident light and chirality of the patterns. Manipulation of the resonantly enhanced SHG and nonlinear emission is facilitated in nanostructures supporting Fano resonances due to controllable intermodal coupling of superradiant and subradiant plasmonic modes, e.g., originating from hybridized dipolar and quadrupolar modes of metamolecules [84]. Their interference leads to asymmetric line shapes observed in the far-field spectra, and it is accompanied by intensified near field stimulating the nonlinear response. Based on near- and far-field radiation properties, specific designs have been suggested for applications in sensing [100,152], shape characterization [153], nonlinear nanorulers [154,155], and nonlinear microscopy [156].

## 3. METAL-DIELECTRIC STRUCTURES

Recently, it was established that very specific nonlinear regimes can be achieved due to magnetic optical response of artificial “meta-atoms” [157]. This picture is at large adapted in the microwave range, and a vast number of research groups are engaged in utilizing tailored nonlinear response of metamaterials, as shown in the recent review [158]. When nonlinearities of both electric and magnetic origin are present, nonlinear response can be modified substantially, being accompanied by nonlinear interference, magnetoelectric coupling, and wave-mixing effects [159,160]. In particular, the interference between the electric and magnetic nonlinear contributions can lead to unidirectional harmonic generation from an ultra-thin metasurface [159] and “non-reciprocal” nonlinear scattering [159] from a symmetric subwavelength nanoelement [161]. The idea of magnetic nonlinearities evolves, being further elaborated by the optics community, and finds implementations in double-layer fishnet structures [162–164,165], coupled metal nanodisks [166,167], and nanostrips [168]. These coupled plasmonic structures sustain antisymmetric oscillations of electron plasma, leading to magnetic-type resonances and the associated enhancement of the nonlinear response. The dissimilar electric and magnetic contributions to the nonlinear response are then disclosed by spectrally resolved measurements and multipolar decomposition of the generated harmonic signal. In Ref. [166], such analysis was performed for SHG observed from a metasurface composed of metal–dielectric–metal nanodisks [see Fig. 3(a)]. The resonant excitation of the magnetic optical mode in composite nanoparticles was shown to govern the efficient nonlinear conversion of the multipolar origin, being a superposition of nonlinearly induced magnetic dipole and electric quadrupole modes with a suppressed contribution from the electric dipole mode. Phenomenological macroscopic description, developed in Ref. [167], established a connection of the resonant enhancement of SHG in such “nanosandwiches” under the excitation of the MD resonance with the dominant role of the nonlinear magnetic-dipole polarization driven by the ${\chi}^{\mathrm{emm}}$ susceptibility. Here, the first index in the superscript corresponds to the electric polarization induced at SH frequency, while the two other denote the magnetic fundamental fields.

Reference [165] reports on the experimental identification of multipolar contributions in the THG signal from a metal–dielectric–metal layered fishnet metamaterial illuminated with a focused pump beam in the vicinity of the magnetic resonance [see Fig. 3(b)]. Previously, the electric and magnetic dipole resonances in fishnets were shown to be responsible for notably distinct THG angular patterns [163]. The metamaterial studied in Ref. [165] consists of $\mathrm{Au}/{\mathrm{MgF}}_{2}/\mathrm{Au}$ structure perforated periodically with an array of holes, and the magnetic response is attained due to antiparallel currents in the top and bottom metal layers. The detected THG radiation pattern, measured by a Fourier imaging technique, was proven to be a result of the interference of the electric dipolar, magnetic dipolar, and electric quadrupolar modes [165].

## 4. ALL-DIELECTRIC NANOSTRUCTURES

Ohmic losses, heating, and low damage thresholds pose substantial restrictions on the achievable performance of the nonlinear plasmonic devices exploiting metallic components, thus limiting their practical use. Nanoparticles made of high-refractive-index dielectrics and semiconductors (such as germanium, tellurium, GaAs, AlGaAs, GaP, and silicon), which do not suffer from large intrinsic absorption at the visible, infrared, and telecom frequencies, are emerging as a promising alternative to plasmonic nanostructures for nanophotonic applications [169,170]. Such high-permittivity nanoparticles exhibit strong interaction with light due to the excitation of the Mie-type resonances they sustain [171–174]. Both these electric and magnetic resonances can be spectrally controlled and engineered independently, and therefore all-dielectric nanostructures offer unique opportunities for the study of nonlinear effects owing to low losses in combination with multipolar optical response of both electric and magnetic nature. Utilizing the Mie-type resonances in resonant subwavelength dielectric geometries has been recently recognized a promising strategy for gaining high efficiencies of nonlinear processes at low mode volumes and designing novel functionalities originating from the optically induced magnetic response [47,175–188]. Particularly, optically magnetic dielectric nanostructures are shown to significantly enhance nonlinear conversion, with high enhancement observed for excitation of the magnetic dipole resonance.

Remarkably, the observation of highly localized fundamental magnetic dipolar mode characterized by circular displacement currents excited in spherical silicon particles of sub-micrometer size in the visible spectrum [189,190] has led to the development of a “magnetic light” concept, which triggers the whole research direction of *all-dielectric nanophotonics* [170,191]. In most previous works on trapped magnetic resonances, silicon was the primary focus of the material [18,33,189,190,192–194] thanks to its CMOS compatibility, low cost, and well-established fabrication methods. Silicon, both crystalline and amorphous, is a centrosymmetric material with inhibited quadratic nonlinearity. However, it possesses a strong bulk third-order optical susceptibility ${\overleftrightarrow{\chi}}_{b}^{(3)}$ [195–203] nearly 200 larger than that of silica along with a moderately high refractive index. Therefore, strong nonlinear phenomena can be anticipated in the optical response of silicon nanoparticles. Figures 4(a)–4(d) present a few examples of the recent experimental study of nonlinear optical properties of silicon-based nanoparticles, their clusters, and metasurfaces.

*Experimental studies of third-harmonic generation.* In the pioneering work [175], it was experimentally confirmed that a strong enhancement of THG can be achieved in isolated silicon nanodisks and their arrays optically pumped in the vicinity of the magnetic dipolar resonance by using THG microscopy and THG spectroscopy techniques. The MD resonance is characterized by the significantly enhanced local fields tightly bound within the volume of nanoscale resonators. Since the induced volume nonlinear source, third-order polarization ${\mathbf{P}}_{\mathrm{bulk}}^{(3\omega )}$, scales with the local electric field ${\mathbf{E}}_{\mathrm{loc}}^{(\omega )}$ cubed, ${\mathbf{P}}_{\mathrm{bulk}}^{(3\omega )}={\u03f5}_{0}{\overleftrightarrow{\chi}}_{b}^{(3)}{\mathbf{E}}_{\mathrm{loc}}^{(\omega )3}$, a considerable enhancement of THG is expected from the nanodisks pumped by the resonant laser radiation. In the experiment, silicon nanodisks placed on silica were subject to an intense femtosecond laser pulse train with frequency close to the magnetic dipole resonance. Analyzing the transmitted signal at TH, it was found that the resonant THG response from silicon nanodisks prevails by a factor of up to 100 over the THG from the bulk silicon slab. The generated 420 nm radiation was bright enough to be observed by a naked eye under the table-lamp illumination conditions [see Fig. 4(a)].

Combining nanodisks into oligomers provides another degree of freedom in tailoring the nonlinear optical response. In particular, strong reshaping of the third-harmonic spectra was observed from trimers of silicon nanodisks with varying interparticle spacings [177]. Complex multipeak patterns in measured THG spectra were substantiated by interference of the nonlinearly generated TH waves radiated by different parts of the sample and augmented by an interplay between the electric and the magnetic dipolar resonances in the nanodisks.

Optical nonlinearities can be further enhanced in finite nanoparticle clusters through the excitation of high-quality collective modes of the magnetic nature. Combining dielectric particles in oligomers may lead to the formation of collective modes with different lifetimes limited by mainly radiative losses, in a striking contrast to deeply subwavelength plasmonic analogues. Such multiresonant dielectric nanostructures can be designed to support high-quality modes of different symmetries, amplifying nonlinear optical effects. In this context, Ref. [185] presents the study of THG from quadrumers composed of four identical silicon nanodisks. Nontrivial wavelength and angular dependencies of the generated harmonic signal were observed, featuring a high enhancement of the nonlinear response associated with a specific type of the magnetic Fano resonance in oligomeric dielectric systems. First predicted in Ref. [204], this Fano resonance arises due to the magneto-electric dipole coupling in individual quadrumers. The measured linear transmission displayed in Fig. 4(b) shows a broad transmission dip and a sharp Fano-like line shape at longer wavelengths; they are attributed to the dominant resonant excitation of different collective modes in the quadrumers. Both resonances are characterized by pronounced magnetic dipolar excitations in the disks and significant local field enhancement and they are accompanied by a substantial increase of the third-harmonic signal, as compared to that from both optically decoupled disks and an unstructured silicon film of the same thickness.

Noticeably, such mechanism for the THG enhancement in a single unit cell of all-dielectric metasurfaces is different from that reported in Ref. [179], where the efficient THG is driven by a high-quality resonance achieved in a two-dimensional array of coupled silicon disks and bars, which stems from collective bright and dark modes of the lattice [see Fig. 4(c)]. In the latter design, nanobars with long axes parallel to the E-polarization of a normally incident radiation and exhibiting electric dipole resonances effectively couple energy from the incident wave into the dark out-of-plane magnetic dipole modes of neighboring silicon disks. A sharp peak in the measured transmittance spectra indicates the Fano interference featuring a resonance with an experimental $Q$-factor of 466. The associated strong enhancement of the local electric field trapped within the disks gives rise to THG amplification factor of $1.5\times {10}^{5}$ with respect to the unpatterned silicon film, leading to conversion efficiency on the order of ${10}^{-6}$, which is the largest value reported for THG on nanoscale using comparable pump energies [179,187].

A simple disk geometry of dielectric nanoparticles appears especially attractive for experimental studies of resonant nonlinear effects because the positions of both their electric and magnetic Mie-type resonances can be easily tuned by varying the aspect ratio [205]. For instance, the efficient excitation of the magnetic dipole mode requires sufficient field retardation throughout the nanodisk to drive the displacement current loop. This suggests that if the particle gets shallow enough, ED and MD resonances become overlapped, gradually expelling the in-plane MD mode. Reference [187] demonstrates that THG in thin Ge nanodisks under normally incident laser excitation can be boosted via the nonradiative anapole mode (AM). This AM is accompanied by the pronounced dip in scattering occurring from destructive far-field interference produced by the electric and toroidal dipole (TD) moments, and its physics can be inferred from the Fano resonance mechanism [206]. The excitation of toroidal multipoles alone attracts much interest fostered by the recent demonstrations of TDs in the engineered metamaterials [207,208]. Since ED and TD have identical scattering patterns, it means that when co-excited and spatially overlapped with the same magnitudes but out of phase, they can cancel the scattering of each other in the far field. This is the basic mechanism behind the recently discovered non-radiating anapoles in individual all-dielectric nanodisks [206]. The AM, which emerges for low-enough aspect ratios, efficiently confines the electric energy inside the disks, which is similar to the performance of MD mode in thicker cylinders. The TH conversion at the anapole mode from a germanium disk array was measured to outperform that at the radiative electric dipolar modes, spectrally surrounding AM, by about one order of magnitude.

*Theoretical approach.* Analytical and numerical analysis of the resonant THG from high-index dielectric nanoparticles was performed in Ref. [47]. This work discusses the basic features and multipolar nature of the nonlinear scattering near the Mie-type optical resonances from simple nanoparticle geometries, such as spheres and disks, with a focus on silicon. Considering a model of a high-permittivity spherical dielectric particle excited in the vicinity of the magnetic dipole resonance, analytical expressions for the electromagnetic field generated at the tripled frequency are derived and it is shown that the TH radiation pattern can be treated as a result of interference of a magnetic dipole and magnetic octupole, which is also confirmed by means of full-wave numerical simulations carried out in COMSOL (see Fig. 5). Besides, in compliance with experimental findings [175,177–179], the efficiency of THG at the MD resonance is noted to significantly exceed that at the ED resonance. Remarkably, due to a larger mode volume of the magnetic Mie mode compared to the electric mode of the same order, the magnetic dipole resonance also superiorly enhances Raman scattering by individual silicon nanospheres [184]. In addition, the approaches for manipulating and directing the resonantly enhanced nonlinear emission by changing shapes and combining dielectric nanoparticles in oligomers are demonstrated [47,185]. This concept is illustrated with an example of a nanoparticle trimer composed of a symmetric dimer of identical silicon spheres placed nearby a smaller nanoparticle [47]. Interparticle interaction influences the multipolar content of the field scattered by a trimer, and the resultant constructive interference between the excited multipoles may provide high directivity of both the linear and nonlinear scattering simultaneously. Alternatively, the preferential backward/forward directionality of THG can be attained for a silicon nanodisk of the adjusted aspect ratio exhibiting spectrally overlapped magnetic and electric dipole resonances [18,47].

*Ultrafast switching.* Silicon nanoparticles employed as nonlinear Mie-type cavities furthermore offer a platform for engineering compact and very speedy photonic switchers. In Ref. [178], the authors distinguish three basic classes of nonlinear phenomena responsible for the optical self-action effect in silicon: (i) Kerr-type processes, including nonlinear refraction and two-photon absorption (TPA); (ii) free carrier generation; and (iii) thermo-optical effect. They were able to design a metasurface of amorphous silicon nanodisks [illustrated in Fig. 4(d)] that suppresses the undesirable free-carrier effects, which are generally slow and pose serious restrictions on the speed of signal conversion, leaving the dominant fast TPA contribution in the nonlinear response. This silicon-based metasurface allowed the achievement of a strong self-modulation of femtosecond pulses with a depth of 60% at picojoule-per-nanodisk pump energies. Pump-probe measurements demonstrated that switching in the nanodisks can be governed by pulse-limited 65 fs long two-photon absorption being enhanced by a factor of 80 with respect to the unstructured silicon film. This result represents an important step toward novel and efficient active photonic devices such as transistors and logic elements to be built in integrated circuits [181]. Such switching speeds potentially allow for the creation of data transmission and processing devices operating at tens and hundreds of terabits per second. Similar to Ref. [178], Kerr and TPA effect-based transmission modulation was reported for a Fano-resonant metasurface in Ref. [179]. Nonlinear light manipulation by scattering diagram and spectra of individual silicon nanoparticles also envisions interesting possibilities for all-optical routing. In Ref. [182] it was demonstrated, both theoretically and experimentally, that ultrafast transient modulation of the dielectric permittivity due to variation of free-carrier (electron-hole plasma) density in silicon, induced via its photoexcitation by femtoseconds laser irradiation, significantly alters scattering behavior of a silicon nanoparticle, supporting a magnetic dipole resonance. Later, accelerated reconfiguration of a radiation pattern from dipole-like to unidirectional (Huygens-source scattering) regime mediated by ultrafast (with times scales of about 2.5 ps) generation of electron-hole plasma during laser pulse–nanoantenna interaction was described in Ref. [186].

*Second-harmonic generation.* For efficient second-order nonlinear applications, excitation of low-order Mie modes in nanoantennas made of high-permittivity noncentrosymmetric semiconductors can be advantageous [180,183,209]. While in plasmonic nanoparticles, the second-order nonlinear response is dominated by surface nonlinearities enhanced by plasmon resonances, in high-index dielectric nanostructures, the bulk nonlinearity may dominate the optical nonlinear interaction in volume, giving rise to better conversion efficiency. In Ref. [180], the disk-shaped nanoantenna made of aluminum gallium arsenide (AlGaAs) is designed for SHG at near-IR wavelengths. The authors consider this choice of material favorable because AlGaAs possesses a large volume quadratic susceptibility, and parasitic TPA effects can be avoided at wavelengths close to 1.55 μm by engineering the Al molar fraction in the alloy composition. The second-order bulk nonlinear susceptibility tensor of AlGaAs, possessing a zinc blended crystalline structure, is anisotropic and contains only off-diagonal elements ${\chi}_{ijk}^{(2)}$ with $i\ne j\ne k$. Thus, in the principal-axis system of the crystal, the $i$th component of the nonlinear source polarization at SH frequency is given by ${P}_{i}^{(2\omega )}={\u03f5}_{0}{\chi}_{ijk}^{(2)}{E}_{j}^{(\omega )}{E}_{k}^{(\omega )}$. Figure 6 shows that the maximum SHG efficiency is achieved for a pump wavelength of 1675 nm, which is close to the magnetic dipole resonance wavelength and predicted to reach values as high as ${10}^{-3}$ at $1\text{\hspace{0.17em}}\mathrm{GW}/{\mathrm{cm}}^{2}$. The conversion efficiency strongly depends on the overlap between the induced nonlinear source and modes generated at SH frequency, and its peak appears slightly shifted from the resonance seen in the linear scattering. The SH nonlinear current distribution resembles an electric quadrupole in this case, and pronounced quadrupolar lobes can be recognized in the multilobe SH far-field radiation pattern [180,183]. In the recent experiment, SH conversion efficiency exceeding ${10}^{-5}$ has been measured for AlGaAs nanocylinders with an optimized geometry pumped at the wavelength of 1.55 μm [188]. Reference [183] further theoretically discusses the shaping SH radiation profile of AlGaAs cylindrical nanoantennas through the engineered multipolar interference by manipulating the pump beam (polarization state, incidence angle, spatial structure) and by changing the disk geometry. In particular, whereas no SH emission can be generated by the AlGaAs nanodisk in the normal forward or backward direction under normal excitation because of the structural symmetry and properties of the AlGaAs nonlinear susceptibility, using a pump beam at tilted incidence is beneficial for obtaining SH signal in the normal directions and increasing the detectable SH power that can be measured through a finite numerical aperture microscope objective.

## 5. CONCLUDING REMARKS AND OUTLOOK

We have discussed multipolar nonlinear effects in resonant metallic, metal–dielectric, and all-dielectric photonic structures. The studies of nonlinear phenomena accompanying the propagation and scattering of light in nanostructured media have been stimulated by a rapid progress in nanofabrication techniques such as electron beam lithography and wet chemical growth, and a growing interest in photonic metamaterials with optically induced magnetic response. The approach based on the multipolar expansion of the electromagnetic fields is closely associated with the Mie scattering theory, and it provides a unified description and useful insights into the physics underlying the nonlinear processes with resonantly driven nanoparticles and nanoparticle clusters. The realization of submicrometer structures of simple geometries exhibiting intense nonlinear response is eminently promising for the optics-based data manipulation and storage technology, sensing applications, and topological photonics [210].

While the extensive theoretical and experimental studies of nonlinear optical properties of metallic nanoparticles laid the foundation of contemporary nonlinear nanoplasmonics, in many cases appealing to the electric dipole resonances, all-dielectric systems have recently been demonstrated to offer stronger nonlinear effects and novel functionalities enabled by pronounced magnetic dipole and higher-order Mie-type resonances. The coexistence of strong electric and magnetic resonances and their interference in high-index dielectric and semiconductor nanostructures bring new physics to simple geometries, opening unique prospects for constructing optical nanoantennas and metasurfaces with reduced dissipative losses and considerable enhancement of the nonlinearly generated fields. We foresee that these latest developments will drive a rapid progress in engineering nonlinear optical effects beyond the diffraction limit, with important implications for novel nonlinear photonic metadevices operating at the nanoscale.

## Funding

Australian Research Council (ARC); Russian Foundation for Basic Research (RFBR) (16-02-00547).

## Acknowledgment

The authors thank M. Kauranen, A. Zayats, H. Giessen, O. J. F. Martin, N. Panoiu, V. Valev, and C. De Angelis for useful discussions and suggestions.

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